On the Structure of Two-Dimensional Constacyclic Codes using Common Zero Sets

TL;DR

This paper characterizes two-dimensional constacyclic codes over finite fields using common zero sets, provides algorithms for their construction and duals, and demonstrates their potential for improved error correction over traditional cyclic codes.

Contribution

It introduces a new characterization method for 2-D constacyclic codes via common zero sets and offers algorithms for their construction and dual code description.

Findings

Codes can have better minimum distance than cyclic counterparts

Algorithm for ideal basis construction from ECZ sets

Parallel encoding scheme demonstrated

Abstract

We consider two-dimensional $(\lambda_1, \lambda_2)$-constacyclic codes over $\mathbb{F}_{q}$ of area $M N$, where $q$ is some power of prime $p$ with $\gcd(M,p)=1$ and $\gcd(N,p)=1$. With the help of common zero (CZ) set, we characterize 2-D constacyclic codes. Further, we provide an algorithm to construct an ideal basis of these codes by using their essential common zero (ECZ) sets. We also describe the dual of 2-D constacyclic codes. Finally, we provide an encoding scheme for generating 2-D constacyclic codes from the generator tensor, implementable in a parallel fashion. Through examples, we illustrate that 2-D constacyclic codes can have better minimum distance compared to their cyclic counterparts with the same code area and code rate, generalizing prior work over 2-D binary cyclic coded arrays.